Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C22 |
| direct product | C2×#1426×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | #367 |
| commutator | C16 | C22×C4×C149 |
| Frattini, cosocle | C2×C16 | C23×C149 |
| prime radical | C23×C149 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ C22×C4×C149 ◁ C22×C8×C149 ◁ C22×C16×C149 ◁ G | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 183 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 3 generators on 183 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183);(4,9,11,22)(5,14)(6,18)(7,19)(10,25)(12,26,29,34)(13,32,30,24)(15,20)(16,28)(17,27)(21,33,31,23);(3,4,10,26,6,12,5,9,19,30,14,23,15,21,25,32,8,11,28,34,20,29,17,22,7,13,27,33,18,31,16,24)7748 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6625);